A Bidirectional Translation Between Analytic Closure Proofs and Law-Level Admissibility

This paper establishes a precise bidirectional translation between analytic closure arguments in partial differential equations and law-level admissibility conditions formulated independently of any specific model.

Building on a sequence of prior works in which closure mechanisms were realized and stress-tested across hostile PDE regimes, the paper abstracts the minimal structural content common to successful analytic proofs. These components include admissible representations under re-expression, a canonized carrier encoding persistence-relevant quantities, an exact drift decomposition separating transfer, persistence, and remainder terms, single-multiplier domination of nonlinear effects, a strict margin or defect-dominance condition yielding coercive contraction, and stability under limits and time localization.

The main result shows that this structural package is sufficient to define a law-level admissibility operator whose fixed points correspond exactly to analytic closure proofs, and conversely, that any admissible law-level object admits at least one faithful realization as an orthodox analytic closure argument. The translation is explicit and reversible, and does not depend on fluid structure, viscosity, incompressibility, or any equation-specific geometry.

Importantly, the paper does not introduce new regularity theorems, dynamical models, or solution concepts. Its contribution is structural and synthetic: it clarifies which elements of analytic reasoning are essential for closure, which are incidental, and how they correspond to a minimal law-level notion of admissibility once equation-specific artifacts are removed.

All arguments are formulated entirely in standard analytic and PDE language. The result provides a formal bridge between concrete proof techniques and abstract closure principles, allowing analytic arguments to be lifted to a law-level description and law-level admissibility statements to be realized concretely, without requiring prior commitment to any particular framework.